$C^0$ Approximations of foliations

Suppose that $\mathcal F$ is a transversely oriented, codimension one foliation of a connected, closed, oriented 3-manifold. Suppose also that $\mathcal F$ has continuous tangent plane field and is {\sl taut}; that is, closed smooth transversals to $\mathcal F$ pass through every point of $M$. We show that if $\mathcal F$ is not the product foliation $S^1\times S^2$, then $\mathcal F$ can be $C^0$ approximated by weakly symplectically fillable, universally tight, contact structures. This extends work of Eliashberg-Thurston on approximations of taut, transversely oriented $C^2$ foliations to the class of foliations that often arise in branched surface constructions of foliations. This allows applications of contact topology and Floer theory beyond the category of $C^2$ foliated spaces.